Rigidity of Einstein manifolds with positive scalar curvature

(1)

DOI 10.1007/s00208-013-0957-7

Mathematische Annalen

Rigidity of Einstein manifolds with positive scalar curvature

Hong-wei Xu · Juan-ru Gu

Received: 1 May 2013 / Revised: 27 June 2013 / Published online: 1 August 2013

Abstract We prove that if Mⁿ(n≥4)is a compact Einstein manifold whose normal- ized scalar curvature and sectional curvature satisfy pinching condition R0> σnKmax, whereσn∈(¹₄,1)is an explicit positive constant depending only on n, then M must be isometric to a spherical space form. Moreover, we prove that if an n(≥4)-dimensional compact Einstein manifold satisfies Kmin ≥ηnR0,whereηn ∈ (¹₄,1)is an explicit positive constant, then M is locally symmetric. It should be emphasized that the pinch- ing constantηnis optimal when n is even. We then obtain some rigidity theorems for Einstein manifolds under(n−2)-th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if M is an n(≥4)-dimensional compact Ein- stein submanifold in the simply connected space form F^N(c)with constant curvature c≥0, and the normalized scalar curvature R0of M satisfies R0> _A_n₊^A_4nⁿ₋₈(c+H²), where An =n³−5n²+8n, and H is the mean curvature of M, then M is isometric to a standard n-sphere.

Mathematics Subject Classification (2000) 53C25·53C20·53C40

Research supported by the NSFC, Grant No. 11071211, 10771187; the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.

H. Xu·J. Gu (

B

⁾

Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China e-mail: [email protected]

H. Xu

e-mail: [email protected]

(2)

1 Introduction

It plays an important role in Riemannian geometry to study the rigidity of Einstein manifolds. To classify the Einstein manifolds satisfying some curvature pinching condition is an important problem, which was initiated by Berger [1]. In 1974, Tachibana [23]

proved that a compact Einstein manifold with positive curvature operator is isometric to a spherical space form. Later, Micallef and Wang [16] proved that a four-dimensional Einstein manifold with nonnegative isotropic curvature is locally symmetric. In 2000, Yang [28] obtained the following rigidity theorem on four-dimensional Einstein manifolds with positive sectional curvature.

Theorem A Let M be a 4-dimensional compact Einstein manifold with Ri cM =3. If the sectional curvature of M satisfies

KM ≥0≡

√1249−23

40 ≈0.308529,

then M is isometric to either the unit 4-sphere S⁴, the 4-dimensional real projective spaceRP⁴, or the complex projective spaceCP².

In 2003, the pinching constant above was improved to 0.292893 by de Araujo Costa in [9]. More discussions about the Einstein manifolds can be seen in [2,13,19], etc.

Recently, Brendle [8] generalized Micallef and Wang’s theorem [16] for 4-dimensional Einstein manifolds to higher dimensional cases.

Theorem B Let M be an n(≥4)-dimensional compact Einstein manifold. If M has positive isotropic curvature, then M is isometric to a spherical space form. Moreover, if M has nonnegative isotropic curvature, then M is locally symmetric.

Let K(π)be the sectional curvature of M for 2-planeπ ⊂TxM. Set Kmax(x):=

max_π⊂T_xM K(π),Kmin(x):=min_π⊂T_xM K(π).Denote by Ri c⁽^k⁾the k-th Ricci cur- vature of M (see Definition2.2below). The following rigidity theorem can be viewed as a consequence of TheoremB.

Theorem C Let M be an n(≥4)-dimensional compact Einstein manifold. If Kmin ≥

1

4Kmax, and the strict inequality holds for some point x0∈M, then M is isometric to a spherical space form.

The purpose of this paper is to prove some new rigidity theorems for Einstein manifolds and submanifolds. In Sect.3, we prove the following rigidity theorem for compact Einstein manifolds with positive scalar curvature.

Theorem 1.1 Let M be an n(≥4)-dimensional compact Einstein manifold. Denote by R0:=c the normalized scalar curvature of M. We have

(i) If R0 > σnKmax,then M is isometric to a spherical space form of constant curvature c.

(3)

(ii) If Kmin ≥ηnR0>0,then M is locally symmetric. In particular, if M is simply connected, then M is isometric to either the standard n-sphere Sⁿ(^√¹_c)or the complex projective spaceCP^m(c)˜ with n=2m.

Here

σn=1− 6

5(n−1), ηn=1− 3

n+2,

˜

c= 4(n−1) n+2 c.

Furthermore, we obtain the following rigidity theorem.

Theorem 1.2 Let M be an n(≥4)-dimensional compact Einstein manifold. Denote by R0 := c and Ri c⁽ⁿ⁻²⁾the normalized scalar curvature and the(n −2)-th Ricci curvature of M. We have

(i) If Ri c⁽_minⁿ⁻²⁾ > τn(n−2)R0,then M is isometric to a spherical space form of constant curvature c.

(ii) If(n−2)R0≥μnRi c⁽maxⁿ⁻²⁾>0, then M is locally symmetric. In particular, if M is simply connected, then M is isometric to either the standard n-sphere Sⁿ(^√¹_c) or the complex projective spaceCP^m(c˜)with n=2m.

Here

τn =1− 6

(n−2)(5n−11),

μn =1− 3

(n−1)(n+1),

˜

c= 4(n−1) n+2 c.

Remark 1.1 We see from Example 3.1that the pinching constants ηn and μn are optimal when n is even.

Let M be an n-dimensional compact submanifold in an N -dimensional Riemannian manifold M^N with mean curvature H . In Sect.4, we extend the theorems above to Einstein submanifolds in a Riemannian manifold with arbitrary codimension, and prove the following rigidity theorem.

Theorem 1.3 Let M be an n(≥4)-dimensional compact Einstein submanifold in the Riemannian manifold M^N. If S≤ ¹⁶₃

Kmin−¹₄Kmax

+ⁿ_n²₋^H₂²,and the strict inequality holds for some point x0∈ M, then M is isometric to a spherical space form.

Remark 1.2 When M is a compact Einstein submanifold of codimension zero, Theo- rem1.3reduces to TheoremC.

(4)

In particular, we obtain the following rigidity theorem for Einstein submanifolds in a space form.

Theorem 1.4 Let M be an n(≥4)-dimensional compact Einstein submanifold in the simply connected space form F^N(c)with constant curvature c. If the normalized scalar curvature R0of M satisfies

R0> An

An+4n−8(c+H²),

where An=n³−5n²+8n, then M is isometric to a spherical space form. Moreover, if c≥0, then M is isometric to a standard n-sphere.

2 Notation and lemmas

Let Mⁿbe an n(≥4)-dimensional submanifold in an N -dimensional Riemannian man- ifold M^N. We shall make use of the following convention on the range of indices.

1≤A,B,C, . . .≤N; 1≤i,j,k, . . .≤n;

if N≥n+1, n+1≤α, β, γ, . . .≤ N.

For an arbitrary fixed point x ∈ M ⊂ M, we choose an orthonormal local frame field{eA}in M^Nsuch that ei’s are tangent to M. Denote by{ωA}the dual frame field of{eA}. Let

Rm=

i,j,k,l

Ri j klωi⊗ωj⊗ωk⊗ωl,

Rm=

A,B,C,D

RA BC DωA⊗ωB⊗ωC⊗ωD

be the Riemannian curvature tensors of M and M, respectively. Denote by h the second fundamental form of M. When N =n,h is identically equal to zero. When N ≥n+1, we set

h=

α,i,j

h_{i j}^αωi ⊗ωj ⊗e_α.

Then we have the Gauss equation

Ri j kl =Ri j kl+ h(ei,ek),h(ej,el) − h(ei,el),h(ej,ek). (2.1) The squared norm S of the second fundamental form of M is given by

S:=

α,i,j

(h^αi j)².

(5)

We put

ξ := 1 n

α,i

h^α_iie_α, H := ξ = 1 n

α

(

i

h^α_ii)².

Definition 2.1 (See also [29], P.349)ξ and H are called the mean curvature vector and mean curvature of M, respectively.

Denote by K(·),K(·),Ri c(·),Ri c(·),R and R the sectional curvatures, the Ricci curvatures and the scalar curvatures of M and M, respectively. Then we have

Ri c(ei)=

j

Ri j i j, Ri c(eA)=

B

RA B A B,

R=

i,j

Ri j i j, R=

A,B

RA B A B.

Set

Kmin(x)= min

π⊂TxMK(π),Kmax(x)= max

π⊂TxMK(π), Kmin(x)= min

π⊂TxM

K(π),Kmax(x)= max

π⊂TxM

K(π).

Then by Berger’s inequality (See e.g. [5], Proposition 1.9), we have

|Ri j kl| ≤ 2

3(Kmax−Kmin) (2.2)

for all distinct indices i,j,k,l, and

|RA BC D| ≤ 2

3(Kmax−Kmin) (2.3)

for all distinct indices A,B,C,D. We set Ri cmin(x)= min

u∈UxMRi c(u), Ri cmin(x)= min

u∈U_xMRi c(u), Ri cmax(x)= max

u∈UxMRi c(u).

For any unit tangent vector u ∈UxM at point x ∈ M,let V_x^kbe a k-dimensional subspace of TxM satisfying u ⊥V_x^k. Choose an orthonormal basis{ei}in TxM such that ej0 =u, span{ej1, . . . ,ejk} =V_x^k, where the indices 1≤ j0,j1, . . . ,jk ≤n are distinct with each other. We set

Ri c⁽^k⁾(u;V_x^k)=Ri c⁽^k⁾([ej₀, . . . ,ej_k])= k q=1

Rj₀j_qj₀j_q. (2.4)

(6)

We extend an orthonormal s-frame {ej₀, . . . ,ej_s₋₁} in TxM to (k +1)-frame {ej₀, . . . ,ej_k}for 1≤s≤k+1≤n and set

R⁽^k^,^s⁾([ej0, . . . ,ejk])=

s−1

p=0

k q=0

Rjpjqjpjq, (2.5)

R⁽^k⁾([ej₀, . . . ,ej_k])=R⁽^k^,^k⁺¹⁾([ej₀, . . . ,ej_k])= k p=0

k q=0

Rj_pj_qj_pj_q. (2.6)

Definition 2.2 We call Ri c⁽^k⁾(u;V_x^k),R⁽^k^,^s⁾([ej₀, . . . ,ej_k]), and R⁽^k⁾([ej₀, . . . ,ej_k]) the k-th Ricci curvature,(k,s)-curvature and k-th scalar curvatrure of M, respectively.

Denote by Ri c⁽_min^k⁾(x),R⁽_min^k^,^s⁾(x)and Ri cmax⁽^k⁾ (x),Rmax⁽^k^,^s⁾(x)the minimum and max- imum of the k-th Ricci curvature and(k,s)-curvature at point x ∈ M for any ortho- normal(k+1)-frame in TxM. Set

R⁽_min^k⁾ =R⁽_min^k^,^k⁺¹⁾, R_max⁽^k⁾ =R_max⁽^k^,^k⁺¹⁾.

The geometry and topology of k-th Ricci curvature was initiated by Hartman [12]

in 1979, and developed by Wu [24] and Shen [20,21], etc.. By the definition above, it is seen that the Ricci curvature of M is equal to the(n−1)-th Ricci curvature and (n−1,1)-curvature; the scalar curvature of M is equal to(n−1,n)-curvature and (n−1)-th scalar curvature. If M is Einstein, then

Ri cmin=Ri cmax= R

n =constant. (2.7)

For any unit tangent vector u ∈UxM at point x ∈ M,let V_x^kbe a k-dimensional subspace of TxM satisfying u⊥V_x^k. Choose an orthonormal basis{eA}in TxM such that eA₀ =u, span{eA₁, . . . ,eA_k} = V_x^k, where the indices 1≤ A0,A1, . . . ,Ak ≤ N are distinct with each other. We define the k-th Ricci curvature as follows.

Ri c⁽^k⁾(u;V_x^k)= k q=1

RA₀AqA₀Aq. (2.8)

Moreover, we define the k-th scalar curvature of M as follows.

R⁽^k⁾([eA0, . . . ,eAk])= k p=0

k q=0

RApAqApAq. (2.9)

Denote by Ri c⁽_min^k⁾(x),R⁽_min^k⁾(x)and Ri c⁽_max^k⁾ (x),R⁽_max^k⁾ (x)the minimum and maxi- mum of the curvatures defined above at point x ∈M.

The following nonexistence theorem for stable currents in a compact Riemannian manifold M isometrically immersed into the simply connected space form F^N(c)is

(7)

employed to eliminate the homology groups Hq(M;Z)for 0< q <n, which was initiated by Lawson-Simons [14] and extended by Xin [25].

Theorem 2.1 Let Mⁿbe a compact submanifold in F^N(c)with c≥0. Assume that n

k=q+1

q i=1

[2|h(ei,ek)|²− h(ei,ei),h(ek,ek)]<q(n−q)c

holds for any orthonormal basis{ei}of TxM at any point x ∈ M, where q is an integer satisfying 0<q <n. Then there are no stable q-currents in M. Moreover,

Hq(M;Z)=Hn−q(M;Z)=0,

where Hi(M;Z) is the i -th homology group of M with integer coefficients, and π1(M)=0 when q =1.

For submanifolds with positive Ricci curvature, we have the following lemma.

Lemma 2.1 [26] Let M be an n(≥4)-dimensional compact submanifold in F^N(c) with c≥0. If the Ricci curvature of M satisfies

Ri cM > n(n−1)

n+2 (c+H²), then M is simply connected.

Proof From Gauss equation, we have Ri c(ei)=(n−1)c+

α,k

[h^α_iih^α_kk−(h^α_{i k})²]. (2.10)

Moreover, we have

−S+n²H²+n(n−1)c=R≥n Ri cmin. This implies that

S−n H²≤n(n−1)(c+H²)−n Ri cmin. (2.11) It follows from (2.10), (2.11) and the assumption that

n k=2

[2|h(e1,ek)|²− h(e1,e1),h(ek,ek)] −(n−1)c

=2

α

n k=2

(h^α_1k)²−

α

n k=2

h^α₁₁h^α_kk−(n−1)c

(8)

=

α

n k=2

(h^α_1k)²−Ri c(e1)

≤ 1

2(S−n H²)−Ri c(e1)

≤ 1

2[n(n−1)(c+H²)−(n+2)Ri cmin]

<0. (2.12)

Hence the assertion follows from Theorem2.1. This proves Lemma2.1.

Lemma 2.2 [11] Let M be a compact Riemannian manifold of dimension n. If M has nonnegative isotropic curvature and has positive isotropic curvature for some point in M, then M admits a metric with positive isotropic curvature.

3 Rigidity theorems for Einstein manifolds

In this section, we will give the proof of Theorems 1.1 and 1.2.

Lemma 3.1 Let M be an n(≥4)-dimensional compact Einstein manifold. Assume M satisfies one of the following conditions:

(i) Ri c⁽_min^k⁾ > (k−⁶₅)Kmaxfor some integer k∈ [2,n−1];

(ii) Ri c⁽_min^k⁾ > ^5k_5k⁻₋⁶₁Ri c⁽max^k⁺¹⁾for some integer k∈ [2,n−2];

(iii) Ri c⁽_min^k⁾ > _5k2^5k+⁻9k⁶−8R⁽max^k⁺¹⁾for some integer k∈ [2,n−2];

(iv) Ri c⁽_min^k⁾ > _s⁽^k₍_5k⁺²2⁾⁽+^5k9k⁻−⁶8⁾)Rmax⁽^k⁺¹^,^s⁾for some integers k ∈ [2,n−2]and s∈ [2,k+2]. Then M is isometric to a spherical space form.

Proof (i) It follows from (2.4) that

Kmin≥Ri c_min⁽^k⁾ −(k−1)Kmax. (3.1) Suppose{e1,e2,e3,e4}is an orthonormal four-frame. It follows from Berger’s inequality (2.2) that

R1234≤ 2

3(Kmax−Kmin).

This together with (3.1) implies that

R1313+R2323+R1414+R2424−2R1234

≥2Ri c_min⁽^k⁾ −2(k−2)Kmax−4

3(Kmax−Kmin)

≥2Ri c_min⁽^k⁾ −2(k−2)Kmax−4

3[k Kmax−Ri c⁽_min^k⁾]

≥ 10 3

Ri c_min⁽^k⁾ − k−6

5

Kmax

. (3.2)

(9)

This together with the assumption implies that M has positive isotropic curvature. It follows from TheoremBthat M is isometric to a spherical space form.

(ii) It’s easy to get from (2.4) that

Kmax≤Ri c_max⁽^k⁺¹⁾−Ri c⁽_min^k⁾. (3.3) This together with the assumption implies that

Ri c⁽_min^k⁾ > 5k−6

5k−1Ri c⁽_max^k⁺¹⁾

≥ 5k−6

5k−1(Kmax+Ri c⁽_min^k⁾).

A direct calculation show that Ri c⁽_min^k⁾ > (k−⁶₅)Kmax,and the conclusion follows from (i).

(iii) By Definition2.2, we obtain Kmax≤ 1

2[Rmax⁽^k⁺¹⁾−(k+3)Ri c⁽_min^k⁾]. (3.4) It follows from (3.4) and the assumption that

Ri c⁽_min^k⁾ > 5k−6

5k²+9k−8R⁽_max^k⁺¹⁾

≥ 5k−6

5k²+9k−8[2Kmax+(k+3)Ri c⁽_min^k⁾].

Then we obtain Ri c_min⁽^k⁾ > (k− ⁶₅)Kmax.This together with (i) implies that M has constant sectional curvature.

(iv) From (2.5) and (2.6), we have Rmax⁽^k⁺¹^,^s⁾

s(k+1) ≥ R⁽max^k⁺¹⁾

(k+1)(k+2), (3.5)

which together with the assumption implies Ri c_min⁽^k⁾ > (k+2)(5k−6)

s(5k²+9k−8)R_max⁽^k⁺¹^,^s⁾

≥ 5k−6

5k²+9k−8R_max⁽^k⁺¹⁾. (3.6) Then the assertion follows from (iii).

This proves the lemma.

(10)

Lemma 3.2 Let M be an n(≥4)-dimensional compact Einstein manifold. Suppose one of the following conditions holds:

(i) Kmin> _k₊¹₃Ri c⁽max^k⁾ for some integer k∈ [2,n−1];

(ii) Ri c⁽_min^k⁺¹⁾ >^k_k⁺₊⁴₃Ri c⁽max^k⁾ for some integer k∈ [2,n−2];

(iii) R_min⁽^k⁺¹⁾> ^k²⁺_k^6k₊₃⁺¹¹Ri cmax⁽^k⁾ for some integer k∈ [2,n−2];

(iv) R_min⁽^k⁺¹^,^s⁾> ^s₍⁽_k^k₊²⁺₂₎₍^6k_k⁺₊¹¹₃₎⁾Ri c⁽max^k⁾ for some integers k∈ [2,n−2]and s∈ [2,k+2].

Then M is isometric to a spherical space form.

Proof (i) From (2.4), we obtain that

Kmax≤ Ri c_max⁽^k⁾ −(k−1)Kmin. (3.7) Suppose{e1,e2,e3,e4}is an orthonormal four-frame. Combing (2.2) and (3.7), we get

R1313+R2323+R1414+R2424−2R1234

≥4Kmin−4

3(Kmax−Kmin)

≥ 16

3 Kmin−4

3[Ri c⁽max^k⁾ −(k−1)Kmin]

≥ 4

3[(k+3)Kmin−Ri c⁽_max^k⁾ ]. (3.8) This together with the assumption implies M has positive isotropic curvature. From TheoremB, we see that M has constant sectional curvature.

(ii) It’s seen from (2.4) that

Kmin≥Ri c_min⁽^k⁺¹⁾−Ri c⁽_max^k⁾ . (3.9) Then we get from the assumption that

Kmin≥ Ri c⁽_min^k⁺¹⁾−Ri c⁽_max^k⁾

> k+4

k+3Ri c_max⁽^k⁾ −Ri c_max⁽^k⁾

= 1

k+3Ri c_max⁽^k⁾ . (3.10)

Therefore, the assertion follows from (i).

(11)

(iii) It follows from (2.4), (2.6) and the assumption that Kmin≥ 1

2[R⁽_min^k⁺¹⁾−(k+3)Ri c⁽max^k⁾ ]

> 1 2

k²+6k+11

k+3 Ri c⁽_max^k⁾ −(k+3)Ri c⁽max^k⁾

= 1

k+3Ri c_max⁽^k⁾ . (3.11)

This together with (i) implies that M has constant sectional curvature.

(iv) By (2.5) and (2.6), we get R_min⁽^k⁺¹^,^s⁾

s(k+1) ≤ R⁽_min^k⁺¹⁾

(k+1)(k+2), (3.12)

which together with the assumption implies Ri c_max⁽^k⁾ < (k+2)(k+3)

s(k²+6k+11)R_min⁽^k⁺¹^,^s⁾

≤ k+3

k²+6k+11R_min⁽^k⁺¹⁾. (3.13) It follows from (iii) that M is isometric to a spherical space form.

This proves the lemma.

Lemma 3.3 Let M be an n(≥4)-dimensional compact Einstein manifold. If one of the following conditions holds:

(i) R_min⁽^k⁾ >

k²+k−²⁴₇

Kmaxfor some integer k∈ [3,n−1];

(ii) R_min⁽^k^,^s⁾ >^s⁽^7k₇²₍⁺_k₊^7k₁⁻₎²⁴⁾Kmaxfor some integers k∈ [3,n−1]and s∈ [2,k+1];

(iii) Kmin> _ks¹₊₆Rmax⁽^k^,^s⁾for some integers k ∈ [1,n−1]and s∈ [2,k+1];

(iv) Kmin> _k2+¹k+6Rmax⁽^k⁾ for some integer k∈ [1,n−1], then M is isometric to a spherical space form.

Proof (i) It follows from (2.6) that

R⁽_min^k⁾ ≤2Kmin+ [k(k+1)−2]Kmax. Then we have

Kmin≥ 1

2[R_min⁽^k⁾ −(k²+k−2)Kmax]. (3.14)

(12)

Suppose{e1,e2,e3,e4}is an orthonormal four-frame. From (2.2), (3.14) and the assumption we get

R1313+R1414+R2323+R2424−2R1234

≥ 1

2{R_min⁽^k⁾ − [k(k+1)−8]Kmax} −4

3(Kmax−Kmin)

≥ 1

2{R_min⁽^k⁾ − [k(k+1)−8]Kmax} −2

3[k(k+1)Kmax−R⁽_min^k⁾]

≥ 7 6

R⁽_min^k⁾ −

k²+k−24 7

Kmax

>0. (3.15)

Therefore, M has positive isotropic curvature. By TheoremB, we see that M has constant sectional curvature.

(ii) By Definition2.2, we have

R⁽_min^k⁾

k(k+1) ≥ R_min⁽^k^,^s⁾

ks . (3.16)

This together with the assumption implies R_min⁽^k⁾ ≥ k+1

s R⁽_min^k^,^s⁾>

k²+k−24 7

Kmax. (3.17)

Then the assertion follows from (i).

(iii) It follows from (2.5) that Kmax≤ 1

2

R_max⁽^k^,^s⁾−(ks−2)Kmin

. (3.18)

Suppose{e1,e2,e3,e4}is an orthonormal four-frame. Then we get from (2.2) and (3.18) that

R1313+R2323+R1414+R2424−2R1234

≥4Kmin−4

3(Kmax−Kmin)

≥ 16

3 Kmin−2

3[Ri c⁽_max^k^,^s⁾−(ks−2)Kmin]

≥ 2

3[(ks+6)Kmin−R⁽_max^k⁾ ]. (3.19) This together with the assumption implies M has positive isotropic curvature. There- fore, M is isometric to a spherical space form.

(iv) By taking s=k+1 in (iii), we get the conclusion.

This completes the proof of Lemma3.3.

(13)

Proof of Theorem 1.1 Since M is Einstein, we know from (2.7) that the normalized scalar curvature and the Ricci curvature satisfy R0= ^{Ri c}_n₋^min₁ = ^{Ri c}_n₋^max₁ .

(i) By taking k = n −1 in conditions (i) of Lemmas3.1, we get that M has constant sectional curvature.

(ii) By taking k = n −1 in (3.8), we see that if Kmin ≥ ⁿ_n⁻₊¹₂R0, then M has nonnegative isotropic curvature. This together with Theorem Bimplies that M is locally symmetric. When M is simply connected, we assume that M is not isometric to the standard n-sphere. Then we claim that for every point x ∈ M there exists an orthonormal four-frame{e1,e2,e3,e4}such that

R1313+R2323+R1414+R2424−2R1234=0. (3.20) Otherwise, M has positive isotropic curvature at some point in M. This together with Lemma2.2implies that M admits a metric with positive isotropic curvature. A result due to Harish [11] says that a compact locally symmetric space which admits a metric of positive isotropic curvature has constant sectional curvature. So M is iso- metric to the standard n-sphere. This is a contradiction. From (3.8), (3.20) and the assumption, we have

Kmin−1

4Kmax≡0, and

Kmin= n−1

n+2R0=n−1

n+2c=constant. (3.21) Therefore, M admits a metric with weakly 1/4-pinched sectional curvature in the global sense. Following Berger’s classification theorem we obtain that M either is homeomorphic to Sⁿor isometric to a compact rank one symmetric space(CROSS).

Since M is locally symmetric, a topological sphere would have to be of constant posi- tive sectional curvature. This contradicts the assumption that M is not isometric to the standard n-sphere. By (3.21) and a simple computation, we know that M is isometric the complex projective spaceCP^m(˜c)with n=2m, wherec˜= ⁴⁽_nⁿ₊⁻₂¹⁾c.

This completes the proof of Theorem1.1.

Proof of Theorem 1.2 (i) By taking k =n−2 in condition (ii) of Lemma3.1, we know that M has constant sectional curvature.

(ii) From

Kmin≥ Ri cmin−Ri c⁽_maxⁿ⁻²⁾, and the assumption

(n−2)R0≥ n²−4

n²−1Ri c_max⁽ⁿ⁻²⁾,

we see that Kmin ≥ ⁿ_n⁻₊¹₂R0and the complex projective space satisfies the equality Kmin=Ri cmin−Ri c⁽maxⁿ⁻²⁾. Hence the assertion follows from (ii) of Theorem1.1.

(14)

This proves Theorem1.2.

From the proof of Theorems 1.1 and 1.2, we have the following corollary.

Corollary 3.1 Let M be an n(≥4)-dimensional compact Einstein Riemannian man- ifold. Denote by R0 := c the normalized scalar curvature of M. Assume one of the following conditions holds:

(i) Kmin> ηnR0;

(ii) (n−2)R0> μnRi c⁽maxⁿ⁻²⁾.

Then M is isometric to a spherical space form of constant curvature c. Hereηn is defined as in Theorem1.1andμnis defined as in Theorem1.2.

By (3.8), (3.9) and a direct calculation, we know that if M is an n(≥4)-dimensional compact manifold satisfies condition (i) or (ii) in Lemma3.2, then

R1313+λ²R1414+R2323+λ²R2424−2λR1234>0

for all orthonormal four-frames{e1,e2,e3,e4}and allλ ∈ R. Using Brendle’s con- vergence result for Ricci flow [4] and taking k=n−1 in condition (i) and k=n−2 in condition (ii), we get the following differentiable sphere theorem.

Theorem 3.1 Let M be an n(≥4)-dimensional compact Riemannian manifold.

Assume one of the following conditions holds:

(i) (n−1)Kmin> ηnRi cmax;

(ii) (n−2)Ri cmin> μn(n−1)Ri c⁽maxⁿ⁻²⁾.

Then M is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to Sⁿ. Hereηnis defined as in Theorem1.1and μnis defined as in Theorem1.2.

For further discussions about the Ricci flow and sphere theorem, we refer to see [3,5–7,10,15,17,18,22,27].

Example 3.1 Let R0 be the normalized scalar curvature of a Riemannian manifold.

By a direct computation, we have the normalized scalar curvatures of the compact rank one symmetric spaces (CROSS) with standard metrics.

R0(CP^m)= m+1

4m−2, dim_R(CP^m)=2m,m≥2;

R0(HP^m)= m+2

4m−1, dim_R(HP^m)=4m,m≥2; R0(OP²)= 3

5, dim_R(OP²)=16.

On the other hand, Kmin(CP^m)=Kmin(HP^m)=Kmin(OP²)= ¹₄.Then we know that the curvatures ofCP^m satisfy

Kmin=η2m

Ri cmax

(2m−1) =η2mR0,

(15)

and

(2m−2)R0=(2m−2)Ri cmin

2m−1 =μ2mRi c⁽_max^2m⁻²⁾.

These mean the pinching constantsηnandμnare optimal when n is even.

Motivated by Theorem1.1and Example3.1, we would like to propose the following conjectures.

Conjecture A Let Mⁿ(n≥4)be a compact Einstein manifold. If R0> ³₅Kmax, then M is isometric to a spherical space form.

Conjecture B Let Mⁿ(n ≥ 4)be an even dimensional compact simply connected Einstein manifold. If KM ≤1 and R0≥cn, where

cn=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ n+2

4(n−1) for n=4 or 4k+2,k∈Z⁺, n+8

4(n−1) for n=4k, k∈Z⁺

[2,∞)and k=4, 3

5 for n=16,

then M is either isometric to the standard n-sphere, or a compact rank one symmetric space.

4 Einstein submanifolds with arbitrary codimension

In this section, we extend the theorems in Sect.3to Einstein submanifolds in a general Riemannian manifold. For compact submanifolds, we first prove Theorem 1.3.

Proof of Theorem 1.3 Setting S_α =_n

i,j=1(h^α_{i j})²,we obtain ⁿ

i=1

h^α_ii 2

=(n−2)

⎡

⎣ⁿ

i=1

(h^α_ii)²+

i=j

(h^α_{i j})²+(_n

i=1h^α_ii)² n−2 −S_α

⎤

⎦. (4.1)

Note that for all distinct p,q,m,l ⁿ

i=1

h^α_ii 2

≤(n−2)

⎡

⎣(h^αpp+h^α_qq)²+(h^αmm+h^α_ll)²+

i=p,q,m,l

(h^α_ii)²

⎤

⎦

=(n−2) _n

i=1

(h^α_ii)²+2h^α_pph^α_qq+2h^α_mmh^α_ll

.

This together with (4.1) implies

2h^α_pph^α_qq+2h^α_mmh^α_ll≥

i=j

(h^αi j)²+(n i=1h^α_ii)²

n−2 −S_α, (4.2)

(16)

for all distinct p,q,m,l. Suppose{e1,e2,e3,e4}is an orthonormal four-frame. From (2.1), we get

R1313+R1414+R2323+R2424−2R1234

=R1313+R1414+R2323+R2424−2R1234

+

α

h^α₁₁h^α₃₃+h^α₂₂h^α₄₄+h^α₂₂h^α₃₃+h^α₁₁h^α₄₄

−(h^α₁₃)²−(h^α₂₃)²−(h^α₂₄)²−(h^α₁₄)²−2(h^α₁₃h^α₂₄−h^α₁₄h^α₂₃)

≥ R1313+R1414+R2323+R2424−2R1234

+

α

h^α₁₁h^α₃₃+h^α₂₂h^α₄₄+h^α₂₂h^α₃₃+h^α₁₁h^α₄₄

−2(h^α13)²−2(h^α23)²−2(h^α24)²−2(h^α14)²

. (4.3)

It follows from Berger’s inequality (2.3) and (4.2) that R1234≤ 2

3(Kmax−Kmin), h^α₁₁h^α₃₃+h^α₂₂h^α₄₄+h^α₂₂h^α₃₃+h^α₁₁h^α₄₄ ≥

i=j

(h^α_{i j})²+(n i=1h^α_ii)² n−2 −S_α.

This together with (4.3) implies that

R1313+R1414+R2323+R2424−2R1234

≥4Kmin−4

3(Kmax−Kmin) +

α i=j

(h_{i j}^α)²+(_n

i=1h^α_ii)² n−2 −S_α

−2(h^α₁₃)²−2(h^α₂₃)²−2(h^α₂₄)²−2(h^α₁₄)²

≥ 16 3

Kmin−1 4Kmax

+n²H²

n−2 −S. (4.4)

Then it follows from the assumption that M has nonnegative isotropic curvature and has positive isotropic curvature for some point x0 ∈ M. This together with Lemma 2.2implies that M admits a metric of positive isotropic curvature. Since M is Ein- stein, it follows from TheoremBthat M is locally symmetric. A result due to Harish [11] says that a compact locally symmetric space which admits a metric of positive isotropic curvature has constant sectional curvature. This completes the proof of

Theorem 1.3.